Quantum ring


Semiconductor Quantum Dots (QDs) [Garcia, 2021] are structures made from the union of two or more semiconductor materials, like or , and are fabricated so their geometry has dimensions in the order of 100 nanometers or less. Due to the difference in energy gaps between each of the semiconductor materials forming the QD, they restrict the mobility of charge carriers like electrons or holes in all three dimensions and, effectively, confine them inside the QD. They exhibit properties that are normally associated with atoms such as quantized energy levels. For this reason QDs are sometimes called artificially fabricated atoms. The nature of the charge carriers confined inside QDs is described by the wavefunction which is obtained by solving the Schrödinger equation.

Quantum rings (QRs) are a type of QD that were first fabricated by Granados et al. [Granados, 2003] in 2003. Since its appearance QRs have been a focus of many condensed matter physicists. And, they have been proven to be useful in many applications like lasers [Cao, 2005], quantum computing [Yu, 2008], single photon emission [Gallardo, 2010], and there are still other studies that show promising qualities for other applications like optoelectronics [Fomin, 2014].

In many of the studies in the literature the authors consider 2-dimensional QRs like the one by Xie [Xie, 2009], but very few consider the 3D nature of the QRs. And this represents one of the advantages that the finite element method has for solving the Schrödinger equation in QDs with rich 3D geometry like QRs.

In this example we will consider a mono-electronic system confined in a quantum ring that is made by the junction of , the material of the ring, and , the surroundings. The objective is to find the eigenstates, which can be the starting point for further calculations like optical absorption, thermodynamical properties, among others.

An electron inside the QR will experience a finite step confinement potential, which means that inside of the ring the potential would be 0, and a finite constant potential outside the ring. In standard conditions of pressure and temperature, the potential outside the QR would be of 256.8 meV, when the concentration aluminum is [Culchac, 2009].


Depiction of the quantum ring and its surroundings. Here the vector represents the position of the electron in the Schrödinger equation.

The behavior of charge carriers inside a crystalline semiconductor, like an electron or a hole, is described by the theory of solid-state physics [Ashcroft, 1976]. In particular the motion of these particles is described by band theory [Zawadzki, 2020]. In this context, a quantity that is analogous to the electron mass is defined. It's called the effective mass and it helps describe how a particle in a periodic potential, such as an electron in a crystal, would respond to external forces, taking into account all the interactions that the particle is summited to inside the crystal. Although, this is a topic with many nuances, see for instance see: [Chang, 2014][Duque-Gomez, 2012].

Generally speaking the effective mass can depend on position and other factors such as temperature and pressure. For this example, as a first approach, we may consider a simplification that is widely used in the literature, which states that for electrons or holes with a small cristal momentum [Zawadzki, 2020], its effective mass can be considered constant. Following the approximation given by Kohn and Sham [Kohn-Sham, 1965], one can assume that every electron will experience the same average potential energy due to the many interactions inside the crystal, such that we can turn a many electron problem into a one-electron Schrödinger equation. Also, we can apply the envelope function approximation [Harrison, 2005], that let us write the time independent Schrödinger equation for the electron's wave function as in equation 1.

The time independent Schrödinger equation for the electron's wave function is given as:

Here represents the effective mass of the electron. For , being the free electron's mass. Also in this case, the potential is a piecewise function that defines a potential energy of outside the ring, and inside it.


We define the ring's mean radius , width , height of the ring , boundary radius , and the height of the whole region , which all represent distances in nanometers:

Specify dimensions of the ring:
Specify dimensions of the bounding domain:

These geometric parameters for this example are inspired by similar theoretical work considering a square cross-section quantum ring [Hernandez, 2022].


We will need to load NDSolve's FEM package.

Load the finite element package:

We create one region for the ring and one for the bounding domain.

Create a ring region with RegionDifference, and the whole domain as a Cylinder:

Then create a region made by the difference of the those two regions above. That way, we are left with a hollow part inside the domain that will define the ring in the mesh later on.

Define a region to be the difference between ring region and the domain region:
Create a boundary mesh from the region with ToBoundaryMesh:

For generating the mesh the "RegionMarker" option is used to set different mesh element sizes in the two different regions, as well as "RegionHoles"->None, such that ring is part of the mesh too.

Generate the mesh from the boundary mesh with ToElementMesh:

Hamiltonian and confinement potential


We define a finite step potential where the potential energy outside the ring is 256.8 meV and 0 inside it. The value of the finite potential is given without units for now, and we will make use of units later when we define the Hamiltonian.

Define a member function with RegionMember:
Define the potential with an Piecewise statement and the Evaluate function:

The SetDelayed statement does not evaluate the right hand side, as a consequence, in every call to the function V the ringMemberFunction will need to be evaluated. That makes the process of solving the deferential equation inefficient. So, the Evaluate function is used in the definition of the potential to make the whole process more efficient. Consider reading this section on efficient evaluation of PDE coefficients on for more details.

Visualize the potential with SliceContourPlot3D:


First, we define the effective mass and the reduced planck constant.

Define reduced planck constant and the effective mass:

Next, we specify the units of the potential with the Quantity function. Since the dimensions of the geometry are in nanometers, we set "ScaleUnits"->{"Meters"->"Nanometers"} so that all units are consistent.

Set up the variables and parameters in vars and pars, respectively:

In the next step we use the Schrödinger PDE component to generate all the PDE terms for us.

Generate the Hamiltonian operator with SchrodingerPDEComponent:

Boundary conditions

We set one Dirichlet condition that implies that the electron's wave function for the low-lying states that we are interested in must decay to 0 at a distance far from the ring. In this case, that is done by setting the wave function to be 0 at the outer boundary of the domain defined earlier.

Use of DirichletCondition function to generate the boundary condition:

Solving the eigenvalue problem

Now, NDEigensystem is used to solve the eigenvalue problem with the Hamiltonian and the mesh previously defined. Where the variables en and funs are defined as the energy eigenvalues and the wave functions, respectively.

Solve the equation and measure the time needed:

It takes less than minute to find eigenstates with a normal laptop.


Let's see what the eigenvalues look like.

Look at the eigenvalues obtained:

To make sense of the units of the eigenvalues we need to know that the SchrodingerPDEComponent changes all units internally to base SI units. And, when we add the specification "ScaleUnits"->{"Meters"->"Nanometers"} in the parameters pars all units of meters are converted to nanometers. So we follow the reverse procedure to convert back to millielectronvolts.

Converted the energies to millielectronvolts, and show them with 5 significant digits:

From this we see that the second and third eigenvalues are basically numerically identical, and the same can be said about the fourth and fifth value. This makes sense since these are degenerate eigenstates. Degenerate eigenvalues are expected in cases where the system's geometry has symmetry, like the rotational symmetry of the quantum ring. This can be more clear by looking at the corresponding probability densities.

Visualize each probability density with DensityPlot3D:

The probability density for the second and third states are basically the same but rotated around the axis. As can be seen from the top view. This is because states and , as well as and , form a degenerate subspace, meaning that any linear combination of those states would lead to a valid solution.

Visualize each function and the boundary mesh from the top:
Visualize the linear combination between states 2 and 3 as well as 4 and 5, in the plane:

Also, one important feature is that by making use of NDEigensystem one obtains the wavefunctions already normalized.

Calculate the integral of the probability density for each state:

On the other hand, is possible to take a closer look to each eigenfunction by doing a contour plot.

Use ContourPlot3D to visualize the wavefunctions with different region functions:

Magnetic field interaction

The effect of an external magnetic field on the eigenstates describing particles confined in low dimensional systems, such as QDs and QRs, is something that has been considered by various authors [Jahan, 2018], [Gutiérrez, 2010]. One of the reasons this is an area of interest is that the interaction with the magnetic field can greatly affect the energy spectrum and consequentially the transition energy between the states confined in the nanostructure. Also, a periodic oscillation in the eigenvalues for each state as the magnetic flux density is increased can be observed, a phenomenon known as Aharanov-Bohm (AB) oscillations. Particularly, AB oscillations are present for particles confined in a QR submitted to the presence of a threading magnetic field [Fomin, 2018]. For this reason we will consider an external constant magnetic field pointing in the direction.

Usually, the Hamiltonian operator for a particle of mass has the following form: , where is the momentum operator that in quantum mechanics is defined as . On the other hand, by following what is known as Peierl's substitution [Luttinger, 1951][Peierls, 1997], when a magnetic field is present the momentum changes from to . Where is the magnetic vector potential, which is defined in such a way that one can obtain the magnetic flux density as .

Therefore, the hamiltonian takes the following form:

We can expand the term as , and therefore, the hamiltonian would take the form in equation 2

Now, applying the definition of , replacing for the effective mass, and for the electron's charge, we can get the PDE operator for the system in question.

One particular characteristic of the magnetic vector potential is that one can add the gradient of a scalar function, for instance , to as , and it would lead us to the same magnetic flux density, since . The function is called gauge function, and choosing a particular and is called gauge fixing. This choice can, in many cases, lead to a simplification of the equations involved. One common way to do this is to use Coulomb's gauge fixing condition, that states that .

As a justification of this condition, we can assume that we have an original vector potential, , which divergence doesn't vanish, i.e, . Then we add to it the gradient of a scalar function such that . And, this way Coulomb's condition, , would lead to the equation . This is a Poisson type equation and it has a unique solution. Therefore, would be uniquely determined by applying the condition . That means, that Coulomb's gauge fixing condition can always be applied without any ambiguity. In summary, since the curl of is specified by , but the divergence of isn't specified, is possible to set the divergence of in a way that let us simplify our equations, in this case .

Considering the Coulomb's gauge fixing condition the PDE operator can be written as is shown in equation 3:

We will consider a magnetic vector potential that satisfies the condition that , where is the position vector, which ensures that is true. Then, we can use the SchrodingerPDEComponent to generate the new PDE operator.

Define the magnetic flux density and the magnetic vector potential:

Note that QuantityMagnitude is used in order to ensure that the SchrodingerPDEComponent can handle correctly the units of .

To generate the PDE operator specify the magnetic vector potential and the charge of the particle in the parameters argument of the SchrodingerPDEComponent function. This way two additional terms are added to the PDE.

Define variables and parameters for the SchrodingerPDEComponent with a magnetic vector potential:
Define the SchrodingerPDEComponent operator:

Note that the Coulomb's gauge for the magnetic vector potential, i.e , is assumed as a default by the SchrodingerPDEComponent.

Solving the eigenvalue problem

We can now proceed to solve the eigenvalue problem, with the same mesh and boundary conditions as before.

Solve the eigenvalue problem and measure the time needed:


The presence of the magnetic field makes the wave functions complex valued, and its necessary to separate the real a imaginary parts of the resulting interpolating function to plot it using the Abs function.

Use ContourPlot3D to visualize the probability densities with a RegionFunction:

The obtained eigenvalues are complex numbers, as the PDE is now complex valued in nature.

Visualize the eigenvalues obtained and convert them to millielectronvolts:

Upon close inspection one can see that the imaginary part is very small compared with the real part, and one can suspect that this is due to numerical noise given the fact that in quantum mechanics the energy eigenvalues are always a real quantity. For this reason we will consider the energy as being the real part of the eigenvalues obtained with NDEigensystem.

If we compare the real part of the eigenvalues when magnetic field is present with when no magnetic field is present, we find that states and , as well as and , are no longer degenerate. Thus, we can say that the presence of the magnetic field removes the degeneracy of the eigenstates we have seen when no magnetic field was present.

Take the difference between the energy eigenvalues obtained in each approach:

Varying the magnetic field

To explore how the external magnetic field affects the energy eigenvalues we can calculate the energy of each state as a function of the magnetic flux density. This can be done by just redefining the PDE operator and solving the eigenvalue problem for each value of the magnetic flux density. For this purpose, a function is defined.

Define a function to get the energy of 5 eigenstates for a given value of the magnetic flux density:

Then Table is used to find the energy for a range of magnetic flux densities.

Find the energies for different magnetic flux densities:

This is a lengthy calculation and takes about minutes to complete with the current mesh in a normal laptop.

Plot the obtained data:

The last graph represents the AB oscillations of the energy levels, as is often referred in the literature. One important feature is that the degeneracy in the eigenstates that can be seen when no magnetic field is present at is revoked by the presence of the magnetic field. The energy levels then begin to split as the magnetic flux density is increased, a phenomenon know as the Zeeman effect. Furthermore, from the graph we can see that the energy of the lowest state starts to grow until a point near where subsequently it starts to fall and then gets back up again at a point near . This oscillation is also present for the remaining states. For instance, the energy of the next state starts decreasing until near when it starts to increase up to a point close to from which it falls back again. And a similar pattern is present for higher states.

In summary, a key feature is that that mere presence of the magnetic field makes the eigenstates no longer degenerate.

On the other hand, something worth noting in the graph is the "anti-crossing" or "avoided crossing" in the energy levels. For instance, take the point close to where the eigenvalues for the first two states come really close and seem to "avoid" each other afterwards. This is a common feature of quantum systems when two eigenvalues won't cross when the strength of a perturbation, as is the of the magnetic field, is increased [Cohen-Tannaoudji, 1992].

To explore the effect that the external field has on the wave functions we can plot the probability densities for a magnetic flux density of . The choice of that specific value will be clear later.

Taking into account the steps in which we varied the magnetic flux density in the previous calculation we can get the wave functions and eigenvalues for a magnetic flux density of .

Get the eigenfunctions from the previous calculation:
Get the eigenvalues from the previous calculation:

This last energies are the real part of the eigenvalues provided by NDEigensystem in millielectronvolts.

Use ContourPlot3D to visualize the probability densities with a RegionFunction:

There are a few things that one can notice here. First, the wave functions for states and have interchange roles compared with the wave functions previously calculated for a magnetic flux density of . And, the same can be said about states and . Moreover, the wave function for state has a morphology that did not appear in the first states for the case. To explain this, its useful to analyze the AB oscillations graph.

Plot the obtained data with GridLines:

We can think about the AB oscillations pattern as a set of displaced parabolas, each representing a state. This way, the energy for a particular eigenstate at , with its correspondent wave function, will evolve following a parabola. In the case of , the probability densities morphologies are the same as in the no magnetic field case, which can be understood by looking at the last figure, and noticing that the ordering of the eigenstates is the same for a value of as for a case. Inversely, for the case with magnetic flux density of the parabolas have intertwined, and consequentially the ordering of the eigenstates has changed, which is reflected in the probability densities as described earlier. Furthermore, state in the case can be traced back, following a parabola, up to the point with a magnetic flux density of , and, as can be seen in the previous graph, in that case it would correspond to state . That is the reason why in the morphology of the probability density for state in the case differs from the previously studied cases.

The choice of the specific magnetic flux density of , is clearer now. Any value between and would let us see how the states have intertwined and how the wave functions have interchanged order due to the magnetic field presence, as can be seen in the graph.

Optical absorption

Once the eigenstates are computed, additional properties, such as the optical absorption can be obtained. To calculate the optical absorption we consider the first two eigenstates obtained for a magnetic flux density of .

We will consider the presence of a magnetic field to do the optical absorption calculation, because in the no-magnetic-field-case states and degenerate, which means that the real physical system will be in a linear combination of the two. Therefore, we will take advantage of the fact that the presence of the magnetic field removes the degeneracy of the eigenstates. And in this case a magnetic flux density of will be sufficient to have the first two eigenstates being non-degenerate.

One way to compute the optical absorption is to follow a procedure described by Karabulut et al. [Karabulut, 2005], that is based on the density matrix formalism and perturbation theory. The details on how the interaction with light is modeled are in Karabulut's paper [Karabulut, 2005]. We can then apply the expressions for the linear optical absorption , the non-linear optical absorption and total optical absorption , that are give in equations 4, 5, and 6 respectively.

Here is the permeability of , and are vacuum and relative electrical permittivity, respectively. The intensity of the incident light beam is and is its frequency. The is the inverse of the decay time between states 1 and 2, due to the interaction between the system and the environment. is the carrier density of states of , is the refractive index and is the speed of light. are the components of the electric dipole matrix defined as = where is the electron's charge and represents the coordinate, which is also in the direction in which the incident light beam is polarized. And finally, is the difference between the two energy levels considered, .

Now we can proceed defining the necessary quantities.

Take out the units for and define the electron's charge:
Define the electric dipole moment matrix = :

The components of the electric dipole matrix must have units of []. But the geometry's and the wave functions's length units are nanometers. So, to have the correct units a factor of is introduced in the integral above. Also, note that is symmetrical.

We set up the other parameters.

Define the parameters needed to calculate the optical absorption:
Define the energy difference without units:

As for the intensity of the light beam and the absorption expressions we will use a similar intensity as the one used in the work by Xie [Xie, 2009].

Defining an intensity:
Define the linear and non-linear absorption functions:

To plot the absorption a factor of is used to have the axis be in meV.

Plot the absorption:
Visualize the electric dipole moment matrix components:
Calculate the resonant frequency finding the maximum in the total absorption:

In the latter plot we can see that the optical absorption has a peak around the transition energy , which means that the system is better able to absorb optical waves with an energy that can promote the ground state to the first excited state. Also, we can see that the absorption has a maximum for a frequency that is in the Terahertz range, and that can be one of the characteristics to consider this type of nanostructure in an optoelectronic application.

Variable effective mass model extension

In the previous approach the effective mass of the electron in the quantum ring was used throughout the whole region. Although this procedure is widely used in the literature, a more appropriate way to solve the problem is to take into account the difference in effective mass in the ring region and the outside region. So, we can write what is commonly know as a BenDaniel-Duke Hamiltonian [BenDaniel, 1966], as in equation 7.

Now, the effective mass is a function of the position, however, still constant in each sub region. Thus, the effective mass will be a piecewise function.

Define the electron's mass:
Redefine the effective mass as a piecewise function:
Visualize the effective mass function:

NDSolve can handle PDE coefficients that are position dependent. The only thing that needs to be done is to make use of the piecewise effective mass in the parameters pars, redefine the magnetic flux density and magnetic vector potential, and generate a new Hamiltonian. The equation can be solved as before with the same geometry and mesh.

Define the magnetic flux density and the magnetic vector potential:

By default a hamiltonian with the diffusion term in the BenDaniel-Duke form is generated.

Redefine the parameters and Hamiltonian:

The boundary condition remains the same as before.

Inspect the previously specified boundary condition:
Find the eigenvalue and vector and measure the time needed in minutes:
Visualize the probability densities:
Convert the energies to the correct units:

It is important to note that the energies have changed and are lower in magnitude than before. To see this more clearly we can plot the energy eigenvalues for each approach.

Plot the energy eigenvalues:
Calculate the energy difference between each approach:
Plot the energy difference between each approach:

Let's explore the difference between the probability densities for the model with constant effective mass with the ones in the BenDaniel-Duke approach.

Calculate the difference in the probability densities between each approach:

Now, let's explore now what happens to the absorption.

Define the energy difference without units:
Redefine the electric dipole moment matrix = , for the new wave functions:
Visualize the new electric dipole moment matrix components:
Plot the absorption:
Calculate the resonant frequency finding the maximum in the total absorption:
Convert the units of the found frequency to Terahertz:

The absorption behaves in a very similar manner as before. And maximum frequency has changed by an amount close to . To compare how the absorption has changed we visualize them.

Plot the different absorption coefficients from each approach:
Plot the different absorption coefficients from each approach:
Plot the different absorption coefficients from each approach:

These graphs illustrate that there are not drastic changes in the absorption. Nevertheless, the absorption coefficients' maxima have decreased and a small redshift can be observed in the peak frequency, when the difference in effective masses between regions is considered.

Take the difference between the two maximum frequencies for each approach:

On the other hand, its also interesting to explore the changes in the eigenstates. By looking at the plots of the wavefunctions for each approach it is difficult to see any difference. Therefore we calculate the integral of the product between an specific eigenstate considering constant effective mass and the same eigenstate considering variable effective mass.

Calculate the integral between solutions with different approaches:

The integrals are close to unity, which is expected when one calculates the integral of a given eigenstate squared, which make one suspect that the eigenfunctions didn't change much. Also, from the plot of the difference in the probability density for each approach is clear that the functional form of the probability density stayed the same.

We can conclude that the changes in the energy eigenvalues are appreciable, close to of the energy of the ground state. The changes in the position of the absorption peak and the maximum absorption value are minor but noticeable. And, this small change can be explained if we take a look at the eigenfunctions: their value and functional form didn't change much. That means that the electric dipole moment matrix components didn't suffered a huge change, and they are a big part of the absorption expressions we have used.

For this problem, we can say that making use of the BenDaniel-Duke Hamiltonian might be dispensable if one is interested in the optical absorption only, but its addition to the model can be essential for calculating the energies properly.


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